Quantum Lyapunov Spectrum

TL;DR

This paper introduces a quantum generalization of Lyapunov exponents, applies it to models like SYK and XXZ, and finds universal random matrix behavior indicative of chaos and entropy production in quantum systems.

Contribution

It presents a novel quantum Lyapunov spectrum, analyzes its statistical properties, and links universality to holographic duality and chaos characterization.

Findings

Black holes are the fastest entropy generators.

Quantum Lyapunov spectrum shows universal random matrix behavior.

Deviations from chaos lead to loss of universality.

Abstract

We introduce a simple quantum generalization of the spectrum of classical Lyapunov exponents. We apply it to the SYK and XXZ models, and study the Lyapunov growth and entropy production. Our numerical results suggest that a black hole is not just the fastest scrambler, but also the fastest entropy generator. We also study the statistical features of the quantum Lyapunov spectrum and find universal random matrix behavior, which resembles the recently-found universality in classical chaos. The random matrix behavior is lost when the system is deformed away from chaos, towards integrability or a many-body localized phase. We propose that quantum systems holographically dual to gravity satisfy this universality in a strong form. We further argue that the quantum Lyapunov spectrum contains important additional information beyond the largest Lyapunov exponent and hence provides us with a better characterization of chaos in quantum systems.